Homogeneous generalized master equation retaining initial correlations

Using the projection operator technique, the exact homogeneous generalized master equation (HGME) for the relevant part of a distribution function (st atistical operator) is derived. The exact (mass) operator governing the evo lution of the relevant part of a distribution function and comprising arbit rary initial correlations is found. Neither the Bogolyubov principle of wea kening of initial correlations with time nor any other approximation such a s random phase approximation has been used to obtain the HGME. These approx imations are usually used to derive the approximate homogeneous equation fo r the relevant part of a distribution function from the conventional exact generalized master equation (GME), which has a source containing the irrele vant part (initial correlations). The HGME does not have a source and conta ins only the linear, relative to the relevant part of a distribution functi on, terms of the GME modified by the dynamics of initial correlations. The obtained equation is valid on any timescale, for any initial moment of time and any initial correlations. In particular, it describes the short-time b ehaviour and allows for treating the influence of initial correlations cons istently. As an example, we have considered a dilute gas of classical parti cles. By selecting the appropriate projection operator, we have derived the homogeneous equation for a one-particle distribution function retaining in itial correlations in the linear approximation on the small density paramet er and for the space homogeneous case. This equation allows for considering all stages of the time evolution. It converts into the conventional Boltzm ann equation on the appropriate timescale if the contribution of all initia l correlations vanishes on this timescale.